2025

Iwaniec, Tadeusz; Onninen, Jani; Radice, Teresa

The concept of complex harmonic potential in a doubly connected condenser (capacitor) is introduced as an analogue of the real-valued potential of an electrostatic vector field. In this analogy the full differential of a complex potential plays the role of the gradient of the scalar potential in the theory of electrostatics. The main objective in the non-static fields is to rule out having the full differential vanish at some points. Nevertheless, there can be critical points where the Jacobian determinant of the differential turns into zero. The latter is in marked contrast to the case of real-valued potentials. Furthermore, the complex electric capacitor also admits an interpretation of the stored energy intensively studied in the theory of hyperelastic deformations. Engineers interested in electrical systems, such as energy storage devises, might also wish to envision complex capacitors as electromagnetic condensers which, generally, store more energy that the electric capacitors.